scholarly journals Harmonic series with polylogarithmic functions

2022 ◽  
Vol 70 (1) ◽  
pp. 43-61
Author(s):  
Vuk Stojiljković ◽  
Nicola Fabiano ◽  
Vesna Šešum-Čavić
Keyword(s):  
Zeta Function ◽  
Harmonic Series ◽  
Harmonic Numbers ◽  
Summation Methods ◽  
Function Series

Introduction/purpose: Some sums of the polylogarithmic function associated with harmonic numbers are established. Methods: The approach is based on using the summation methods. Results: This paper generalizes the results of the zeta function series associated with the harmonic numbers. Conclusions: Various interesting series as the consequence of the generalization are obtained.

scholarly journals LOG-TANGENT INTEGRALS AND THE RIEMANN ZETA FUNCTION

2019 ◽  
Vol 24 (3) ◽  
pp. 404-421
Author(s):  
Lahoucine Elaissaoui ◽  
Zine El-Abidine Guennoun
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Integrable Function ◽  
Harmonic Series ◽  
Algebraic Properties ◽  
Tangent Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Square Integrable Function ◽  
Positive Integers

We show that integrals involving the log-tangent function, with respect to any square-integrable function on (0,π/2), can be evaluated by the harmonic series. Consequently, several formulas and algebraic properties of the Riemann zeta function at odd positive integers are discussed. Furthermore, we show among other things, that the log-tangent integral with respect to the Hurwitz zeta function defines a meromorphic function and its values depend on the Dirichlet series ζh(s) :=∑n≥1hnn−s−8, where hn=∑nk=1(2k−1)−1.

scholarly journals Why the Glove of Mathematics Fits the Hand of the Natural Sciences So Well :How Far Down the (Fibonacci) Rabbit Hole Goes

2016 ◽  
Vol 12 (15) ◽  
pp. 1 ◽  
Author(s):  
David F. Haight
Keyword(s):  
Zeta Function ◽  
Good Reason ◽  
Fibonacci Sequence ◽  
Prime Numbers ◽  
Natural Sciences ◽  
Harmonic Series ◽  
General Question ◽  
Isaac Newton ◽  
Preestablished Harmony ◽  
The One

Why does the glove of mathematics fit the hand of the natural sciences so well? Is there a good reason for the good fit? Does it have anything to do with the mystery number of physics or the Fibonacci sequence and the golden proportion? Is there a connection between this mystery (golden) number and Leibniz’s general question, why is there something (one) rather than nothing (zero)? The acclaimed mathematician G.H. Hardy (1877-1947) once observed: “In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy.” Is this also true of great physics? If so, is there a simple “preestablished harmony” or linchpin between their respective ultimate foundations? The philosopher-mathematician, Gottfried Leibniz, who coined this phrase, believed that he had found that common foundation in calculus, a methodology he independently discovered along with Isaac Newton. But what is the source of the harmonic series of the natural log that is the basis of calculus and also Bernhard Riemann’s harmonic zeta function for prime numbers? On the occasion of the three-hundredth anniversary of Leibniz’s death and the one hundredth-fiftieth anniversary of the death of Bernhard Riemann, this essay is a tribute to Leibniz’s quest and questions in view of subsequent discoveries in mathematics and physics. (In the Journal of Interdisciplinary Mathematics, Dec. 2008 and Oct. 2010, I have already sympathetically discussed in detail Riemann’s hypothesis and the zeta function in relation to primes and the zeta zeros. Both papers were republished online in 2013 by Taylor and Francis Scientific Publishers Group.)

A generalized alternating harmonic series

2021 ◽  
Vol 6 (12) ◽  
pp. 13480-13487
Author(s):  
Shelby Kilmer ◽  
◽  
Songfeng Zheng
Keyword(s):  
Harmonic Series ◽  
Harmonic Numbers ◽  
The Relationship

<abstract><p>This paper introduces a generalization of the alternating harmonic series, expresses the sum in two closed forms, and examines the relationship between these sums and the harmonic numbers.</p></abstract>

scholarly journals Explicit evaluation of quadratic Euler sums

2017 ◽  
Vol 13 (03) ◽  
pp. 655-672 ◽  
Author(s):  
Ce Xu ◽  
Yingyue Yang ◽  
Jianwen Zhang
Keyword(s):  
Zeta Function ◽  
Closed Form ◽  
Riemann Zeta Function ◽  
Harmonic Numbers ◽  
Euler Sums ◽  
Explicit Evaluation ◽  
Explicit Formulae ◽  
Riemann Zeta ◽  
The Given ◽  
Double Nonlinear

In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. As applications of these formulae, we give new closed form representations of several quadratic Euler sums through Riemann zeta function and linear sums. The given representations are new.

scholarly journals A generalized alternating harmonic series

2021 ◽  
Vol 6 (12) ◽  
pp. 13480-13487
Author(s):  
Shelby Kilmer ◽  
◽  
Songfeng Zheng
Keyword(s):  
Harmonic Series ◽  
Harmonic Numbers ◽  
The Relationship

<abstract><p>This paper introduces a generalization of the alternating harmonic series, expresses the sum in two closed forms, and examines the relationship between these sums and the harmonic numbers.</p></abstract>

A Zeta Function Series: 10786

2001 ◽  
Vol 108 (1) ◽  
pp. 84
Author(s):  
Leroy Quet ◽  
Reiner Martin ◽  
Otto G. Ruehr
Keyword(s):  
Zeta Function ◽  
Function Series

The Riemann zeta-function and its derivatives

1995 ◽  
Vol 450 (1940) ◽  
pp. 477-499 ◽  
Keyword(s):  
Zeta Function ◽  
Error Bounds ◽  
Riemann Zeta Function ◽  
Summation Methods ◽  
Stieltjes Constants ◽  
The Riemann Zeta Function ◽  
Higher Derivatives ◽  
Riemann Zeta ◽  
Significant Figures ◽  
Derivatives Of

Formulas for higher derivatives of the Riemann zeta-function are developed from Ramanujan’s theory of the ‘constant’ of series. By using the Euler-Maclaurin summation methods, formulas for ζ( n )( s ), ζ( n )(1 – s ) and ζ( n )(0) are obtained. Additional formulas involving the Stieltjes constants are also derived. Analytical expression for error bounds is given in each case. The formulas permit accurate derivative evaluation and the error bounds are shown to be realistic. A table of ζ'( s ) is presented to 20 significant figures for s = –20(0.1)20. For rational arguments, ζ(1/ k ), ζ'(1/ k ) are given for k = –10(1)10. The first ten zeros of ζ'( s ) are also tabulated. Because the Stieltjes constants appear in many formulas, the constants were evaluated freshly for this work. Formulas for the γ n are derived with new error bounds, and a tabulation of the constants is given from n = 0 to 100.

scholarly journals POWER SERIES WITH SKEW-HARMONIC NUMBERS, DILOGARITHMS, AND DOUBLE INTEGRALS

2013 ◽  
Vol 56 (1) ◽  
pp. 93-108
Author(s):  
Khristo N. Boyadzhiev
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Harmonic Series ◽  
Partial Sums ◽  
Simultaneous Solution ◽  
College Mathematics ◽  
Harmonic Numbers ◽  
Numerical Series ◽  
Double Integrals

ABSTRACT The skew-harmonic numbers are the partial sums of the alternating harmonic series, i.e., the expansion of log 2.We evaluate in closed form various power series and numerical series with skew-harmonic numbers. This provides a simultaneous solution of two recent problems by Ovidiu Furdui in the American Mathematical Monthly and the College Mathematics Journal. We also present and discuss representations involving the dilogarithm and the trilogarithm which are related to our results. Finally, we provide the evaluations of several double integrals in terms of classical constants.

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